A092766 Triangle read by rows: coefficients of Yablonskii-Vorob'ev polynomials.
1, 1, 1, -1, 1, -5, -5, 1, -15, 0, -175, 1, -35, 175, -1225, -12250, 6125, 1, -70, 1155, -9800, -67375, -1414875, 4716250, 2358125, 1, -126, 4725, -80850, 242550, -12733875, -202327125, 3034906875, 0, 11802415625, 1, -210, 15015, -512050, 7882875, -121396275, -1618617000, -24886236375, 1933235679375, -6750981737500, 35442654121875, 177213270609375, -59071090203125
Offset: 0
Examples
T(0) = 1, T(1) = x, T(2) = x^3 - 1, T(3) = x^6 - 5*x^3 - 5, T(4) = x^10 - 15*x^7 - 175*x, T(5) = x^15 - 35*x^12 + 175*x^9 - 1225*x^6 - 12250*x^3 + 6125, ... From _Gheorghe Coserea_, Nov 10 2016: (Start) Triangle starts: n\k [0] [1] [2] [3] [4] [5] [0] 1; [1] 1; [2] 1, -1; [3] 1, -5, -5; [4] 1, -15, 0, -175; [5] 1, -35, 175, -1225, -12250, 6125; ... (End)
Links
- Gheorghe Coserea, Rows n = 0..44, flattened
- M. Kaneko and H. Ochiai, On coefficients of Yablonskii-Vorob'ev polynomials
Programs
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Mathematica
T[0][] = 1; T[1][x] := x; T[n_][x_] := T[n][x] = (x T[n-1][x]^2 + T[n-1][x] T[n-1]''[x] - T[n-1]'[x]^2)/T[n-2][x] // Simplify; row[n_] := Join[{1}, Partition[CoefficientList[T[n][x], x] // Reverse // Rest, 3][[All, 3]]]; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Oct 23 2018 *)
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PARI
T(n)=if(n<2,if(n<1,n>=0,x),(x*T(n-1)^2+T(n-1)*T(n-1)''-T(n-1)'^2)/T(n-2))
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PARI
seq(N) = { my(x = 'x, t = vector(N)); t[1] = x; t[2] = x^3 - 1; for (n = 3, N, t[n] = (x*t[n-1]^2 + t[n-1]*t[n-1]'' - t[n-1]'^2)/t[n-2]); concat(1, t); }; pol2row(p) = { my(tn = poldegree(p)); vector(1 + tn\3, k, polcoeff(p, tn-3*(k-1))); }; concat(apply(pol2row, seq(8))) \\ Gheorghe Coserea, Nov 10 2016
Formula
T(n) = Sum {k = 0..A008748(n)-1} a(n,k) * x^(A000217(n) - 3*k), where T(n)*T(n-2) = x*T(n-1)^2 + T(n-1)*T(n-1)'' - T(n-1)'^2, with T(0) = 1, T(1) = x. - Gheorghe Coserea, Nov 10 2016
Extensions
Offset corrected by Gheorghe Coserea, Nov 10 2016
Comments